Minimal Spanning Trees for Graphs with Random Edge Lengths

  • J. Michael Steele
Conference paper
Part of the Trends in Mathematics book series (TM)


The theory of the minimal spanning tree (MST) of a connected graph whose edges are assigned lengths according to independent identically distributed random variables is developed from two directions. First, it is shown how the Tutte polynomial for a connected graph can be used to provide an exact formula for the length of the minimal spanning tree under the model of uniformly distributed edge lengths. Second, it is shown how the theory of local weak convergence provides a systematic approach to the asymptotic theory of the length of the MST and related power sums. Consequences of these investigations include (1) the exact rational determination of the expected length of the MST for the complete graph Kn for 2 ≤ n ≤ 9 and (2) refinements of the results of Penrose (1998) for the MST of the d-cube and results of Beveridge, Frieze, and McDiarmid (1998) and Frieze, Ruzink6, and Thoma (2000) for graphs with modest expansion properties. In most cases, the results reviewed here have not reached their final form, and they should be viewed as part of work-in-progress.


Poisson Process Connected Graph Edge Length Minimal Span Tree Geometric Graph 
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  1. [1]
    D.J. Aldous (1992) Asymptotics in the random assignment problem Probab. Th. Rel. Fields, 93, 507–534.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    D.J. Aldous (2001) The ç(2) limit in the random assignment problem Random Structures Algorithms, 18, 381–418.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    D.J. Aldous and J.M. Steele (2002) The Objective Method and the Theory of Local Weak Convergence in Discrete and Combinatorial Probability (ed. H. Kesten), Springer-Verlag, New York. [in press]Google Scholar
  4. [4]
    N. Alon, A. Frieze, and D. Welsh (1994) Polynomial time randomized approximation schemes for the Tutte polynomial of dense graphs pp. 24–35 in the Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (S. Goldwasser, ed.), IEEE Computer Society Press, 1994.Google Scholar
  5. [5]
    F. Avram and D. Bertsimas (1992) The minimum spanning tree constant in geometric probability and under the independent model: a unified approach Annals of Applied Probability 2, 113–130.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    A. Beveridge, A. Frieze, C. McDiarmid (1998) Minimum length spanning trees in regular graphs Combinatorica 18, 311–333.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    R.M. Dudley (1989) Real Analysis and Probability Wadworth Publishing, Pacific Grove CA.zbMATHGoogle Scholar
  8. [8]
    A.M. Frieze (1985) On the value of a random minimum spanning tree problem Discrete Appl. Math. 10, 47–56.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    A.M. Frieze and C.J.H. McDiarmid (1989) On random minimum lenght spanning trees Combinatorica, 9, 363–374.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    A. M. Frieze, M. Ruszinkó, and L. Thoma (2000) A note on random minimum lenght spanning trees Electronic Journal of Combinatorics 7, Research Paper 5, (5 pp.)Google Scholar
  11. [11]
    I.M. Gessel and B.E. Sagan (1996) The Tutte polynomial of a graph depth-first search and simpicial complex partitions Electronic Journal of Combinatorics 3, Reseach Paper 9, (36 pp.)Google Scholar
  12. [12]
    T.H. Harris (1989) The Theory of Branching Processes Dover Publications, New York.Google Scholar
  13. [13]
    S. Janson (1995) The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph Random Structures and Algorithms 7, 337–355.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    D. Karger (1999) A randomized fully polynomial time approximation scheme for the all-terminal network reliablity problem SIAM Journal of Computing 29, 492–514.MathSciNetCrossRefGoogle Scholar
  15. [15]
    J.F.C. Kingman (1993) Poisson Processes Oxford Universtiy Press, New York, 1993.zbMATHGoogle Scholar
  16. [16]
    P.A.P. Moran (1967) A non-Markovian quasi-Poisson process Studia Scientiarum Mathematicarum Hungarica 2, 425–429.MathSciNetzbMATHGoogle Scholar
  17. [17]
    S. Negami (1987) Polynomial Invariants of Graphs Transactions of the American Mathematical Society 299, 601–622.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    G. Parisi (1998) A conjecture on radon’ bipartite matching ArXiv Condmat 980–1176.Google Scholar
  19. [19]
    M. Penrose (1998) Random minimum spanning tree and percolation on the n-cube Random Structures and Algorithms 12, 63–82.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    A. Rényi, (1967) Remarks on the Poisson Process Studia Scientiarum Mathematicarum Hungarica 2, 119–123.MathSciNetzbMATHGoogle Scholar
  21. [21]
    J.M. Steele(1987) On Frieze’s ζ(3)limit for lengths of minimal spanning trees Discrete Applied Mathematics, 18 (1987), 99–103.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    J.M. Steele (1997) Probability Theory and Combinatorial Optimization NSFCBMS Volume 69. Society for Industrial and Applied Mathematics, Philadelphia.CrossRefGoogle Scholar
  23. [23]
    D. Welsh (1999) The Tutte Polynomial Random Structures and Algorithms 15, 210–228.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    D. Williams (1991) Probability with Martingales Cambridge University Press, Cambridge.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • J. Michael Steele
    • 1
  1. 1.Department of Statistics Wharton SchoolUniversity of PennsylvaniaPhiladelphia

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